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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblem 3
Chapter 6, Problem 3

Find the value of the objective function at each corner of the graphed region. What is the maximum value of the objective function? What is the minimum value of the objective function? 1. Objective Function z=40x+50y
Graph showing a shaded polygon with vertices at (0,0), (0,8), (4,9), and (8,0) on an xy-coordinate plane.

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Identify the corner points of the feasible region from the graph. The points are (0, 0), (0, 10), (6, 8), and (10, 0).
Write down the objective function: \(z = 40x + 50y\).
Calculate the value of the objective function at each corner point by substituting the coordinates into the function:
At (0, 0): \(z = 40(0) + 50(0)\)
At (0, 10): \(z = 40(0) + 50(10)\)
At (6, 8): \(z = 40(6) + 50(8)\)
At (10, 0): \(z = 40(10) + 50(0)\)
Compare the calculated values to determine which is the maximum and which is the minimum value of the objective function.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Objective Function

An objective function is a mathematical expression that defines the goal of an optimization problem, often to maximize or minimize a value. In this problem, z = 40x + 50y represents the objective function, where x and y are variables, and the goal is to find the maximum and minimum values of z within the feasible region.
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Feasible Region and Corner Points

The feasible region is the set of all possible points (x, y) that satisfy the problem's constraints, shown as the shaded area on the graph. The corner points (vertices) of this region are critical because, according to the linear programming theory, the maximum and minimum values of the objective function occur at these points.
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Evaluating the Objective Function at Corner Points

To find the maximum and minimum values of the objective function, substitute the coordinates of each corner point into the function z = 40x + 50y. Calculate z for each vertex, then compare these values to identify which is the largest (maximum) and which is the smallest (minimum).
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Related Practice
Textbook Question

In Exercises 1–4, determine whether the given ordered pair is a solution of the system. (2, 5) 2x + 3y = 17 x + 4y = 16

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Textbook Question

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.

Textbook Question

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. 6x2−14x−27(x+2)(x−3)2\(\frac{6x^2-14x-27}{\left(x+2\right)(x-3)^2}\)

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Textbook Question

In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. (3x+16)/(x + 1) (x − 2)²

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Textbook Question

Graph each inequality. x−2y>10

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Textbook Question

In Exercises 1–18, solve each system by the substitution method. {y=x2−4x−10y=−x2−2x+14\(\begin{cases}\)y = x^2 - 4x - 10 \(\y\) = -x^2 - 2x + 14\(\end{cases}\)